g-frame representations with bounded operators

Abstract

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form {T i f 1 } i∈N , where f 1 belongs to Hilbert space H and T : H → H belongs to certain classes of the bounded operators. Christensen et al., study frames for H with index set N (or Z), that have representations in the formAs frames of subspaces, fusion frames and generalized translation invariant systems are the spacial cases of g-frames, the purpose of this paper is to study gframes Λ = {Λ i ∈ B(H, K) : i ∈ I} (I = N or Z) having the form Λ i+1 = Λ 1 T i , for T ∈ B(H).

Properties of bounded representations for $G$-frames

“…Sequence Λ = {Λ i ∈ B(H, K i ) : i ∈ N} is a g-frame (respectively, g-Bessel family, g-Riesz basis, g-orthonormal basis) if and only if {Λ * i e i,j } i∈N,j∈J i is a frame (respectively, Bessel sequence, Riesz basis, orthonormal basis). Now we summarize some results of article [18] in which we generalize the results of articles [10,11] to introduce the representation of g-frames with bounded operators.…”